On the?-Sylvester Equation AX ±X?B? = C Chun-Yueh Chiang ∗ Eric King-wah Chu † Wen-Wei Lin‡

We consider the solution of the?-Sylvester equations AX±X?B? = C, for? = T,H and A,B, ∈ Cn×n, and the related linear matrix equations AXB?±X? = C, AXB?±CX?D? = E and AX±X?A? = C. Solvability conditions and stable numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize on the square cases where m = n but the rectangular cases will be considered